The non-equilibrium stress in the current configuration in a Maxwell branch writes (trapezoidal integration) $$ \begin{align*} \mathbf{h}_j^{n+1} \approx e^{-\frac{\Delta t}{\tau_j}} \frac{1}{\Delta J} \Delta F ~\mathbf{h}_j^{n}(\Delta F)^T + \Gamma_j \frac{1 - e^{-\frac{\Delta t}{\tau_j}}}{\frac{\Delta t}{\tau_j}}\left[ \boldsymbol{\sigma}^{n+1}_0 - \frac{1}{\Delta J} \Delta F ~~\boldsymbol{\sigma}^{n}_0(\Delta F)^T\right] \end{align*} $$ where $\Delta \mathbf{F} = \mathbf{F}^{n+1}\left(\mathbf{F}^{n}\right)^{-1}$ and $\Delta J = \text{det}\left(\Delta \mathbf{F}\right)$.

The nonlinear Maxwell brach is composed of a nonlinear spring (deviatoric $\psi_e$ and volumetric $\psi_{vol}$ parts) and a dashpot ($\dot{\gamma}$).

Within the branch, the total deformation gradient can be multiplicatively decomposed as $$ \mathbf{F} = \mathbf{F}^e\mathbf{F}^v \rightarrow \mathbf{F}^e = \mathbf{F}\left(\mathbf{F}^v\right)^{-1} $$ and the time-derivative of the viscous deformation gradient writes $$ \dot{\mathbf{F}}^v = \dot{\gamma}\frac{\text{dev}\left(\boldsymbol{\sigma}^e\right)}{\sqrt{\text{dev}\left(\boldsymbol{\sigma}^e\right):\text{dev}\left(\boldsymbol{\sigma}^e\right)}}\mathbf{F}^v = \dot{\gamma}\frac{\text{dev}\left(\boldsymbol{\sigma}^e\right)}{\tau}\mathbf{F}^v = \dot{\gamma}\mathbf{N}\mathbf{F}^v, $$ which allows computing the time-evolution of the viscous deformation gradient $\mathbf{F}^v$.

The integration of $\mathbf{F}^v$ is performed using an exponential map integrator (read this document for more details), either explicitely $$ \mathbf{F}^v_{n+1} = \text{exp}\left[\Delta t \dot{\gamma}_{n} \mathbf{N}_{n}\right]\mathbf{F}^v_{n} $$ or implicitely $$ \mathbf{F}^v_{n+1} = \text{exp}\left[\Delta t \dot{\gamma}_{n+1} \mathbf{N}_{n+1}\right]\mathbf{F}^v_{n} $$ using a local Newton-Raphson iteration scheme.

The deviatoric (elastic) potential $\psi_{e}^{(i)}$ is defined using hyperelastic potential laws defined in Deviatoric Potentials whilst volumetric potential $\psi_{vol}^{(i)}$ is defined using volumic potential laws in Volumic Potentials.

The creep factor $\dot{\gamma}$ is defined using dashpot laws defined in hyper_vol_dashpot.